# Browsing by Subject "Wave equation"

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Item Fast parallel solution of heterogeneous 3D time-harmonic wave equations(2012-12) Poulson, Jack Lesly; Ying, Lexing; Engquist, Bjorn; Fomel, Sergey; Ghattas, Omar; van de Geijn, RobertShow more Several advancements related to the solution of 3D time-harmonic wave equations are presented, especially in the context of a parallel moving-PML sweeping preconditioner for problems without large-scale resonances. The main contribution of this dissertation is the introduction of an efficient parallel sweeping preconditioner and its subsequent application to several challenging velocity models. For instance, 3D seismic problems approaching a billion degrees of freedom have been solved in just a few minutes using several thousand processors. The setup and application costs of the sequential algorithm were also respectively refined to O(γ^2 N^(4/3)) and O(γ N log N), where N denotes the total number of degrees of freedom in the 3D volume and γ(ω) denotes the modestly frequency-dependent number of grid points per Perfectly Matched Layer discretization. Furthermore, high-performance parallel algorithms are proposed for performing multifrontal triangular solves with many right-hand sides, and a custom compression scheme is introduced which builds upon the translation invariance of free-space Green’s functions in order to justify the replacement of each dense matrix within a certain modified multifrontal method with the sum of a small number of Kronecker products. For the sake of reproducibility, every algorithm exercised within this dissertation is made available as part of the open source packages Clique and Parallel Sweeping Preconditioner (PSP).Show more Item Modeling of wave phenomena in heterogeneous elastic solids(2003-05) Romkes, Albert; Oden, J. Tinsley (John Tinsley), 1936-Show more This dissertation addresses the analysis of the classical problem in con tinuum mechanics of wave propagation through heterogeneous elastic media. The class of waves that are considered are stress waves propagating through linearly elastic media with highly oscillatory material properties. This work provides an approach which resolves a classical open problem: the accurate characterization of interfacial stresses in highly heterogeneous media through which stress waves propagate. This is accomplished using an extension of the theory and methodology of adaptive modeling to complex sesquilinear forms. A general abstract notion of residual based a posteriori error analysis is presented, which makes possible the development of a mathematical framework for the mathematical modeling and numerical analysis of this elastodynamic problem. viii The notion of hierarchical modeling is ﬁrst applied to the derivation of computable and reliable estimates of the modeling error in a speciﬁc quantity of interest: the average stress on a subdomain in the elastic body. The estimate is subsequently employed in a goal-oriented adaptive modeling algorithm that is introduced for solving wave propagation in heterogeneous media. To control the error due to geometric dispersion, the algorithm solves the wave problem in the complex frequency domain by iteratively adapting the mathematical material model until the error estimate meets a preset tolerance. The algo rithm is applicable to elastic materials with arbitrary microstructure and does not require geometric periodicity. A number of one-dimensional steady-state and transient examples are investigated, which demonstrate the application of an adaptive modeling algorithm and the reliability and accuracy of the error estimate. A new Discontinuous Galerkin Method (DGM) is presented to numer ically solve the wave equation in the frequency domain. Well-posedness and convergence of the formulation is proved for the case of a Reaction-Diﬀusion type model problem. One- and two-dimensional numerical veriﬁcations are shown. The general abstract framework of a posteriori error analysis is then again applied, but now to the new DGM formulation of the wave equation to derive an estimate of the numerical approximation error in the quantity of in terest. An hp-adaptive algorithm for numerical error control is introduced and numerical results are presented for one-dimensional steady state applications.Show more Item Weakly non-local arbitrarily-shaped absorbing boundary conditions for acoustics and elastodynamics theory and numerical experiments(2004) Lee, Sanghoon; Kallivokas, Loukas F.Show more In this dissertation we discuss the performance of a family of local and weakly non-local in space and time absorbing boundary conditions, prescribed on trun cation boundaries of elliptical and ellipsoidal shape for the solution of two- and three-dimensional scalar wave equations, respectively, in both the time- and frequency-domains. The elliptical and ellipsoidal artiﬁcial boundaries are de rived as particular cases of general arbitrarily-shaped convex boundaries for which the absorbing conditions are developed. From the mathematical per spective, the development of the conditions is based on earlier work by Kalli vokas et al [72–77]; herein an incremental modiﬁcation is made to allow for the spatial variability of the conditions’ absorption characteristics. From the appli cations perspective, the obtained numerical results appear herein for the ﬁrst time. It is further shown that the conditions, via an operator-splitting scheme, lend themselves to easy incorporation in a variational form that, in turn, leads to a standard Galerkin ﬁnite element approach. The resulting wave absorbing ﬁnite elements are shown to preserve the sparsity and symmetry of standard ﬁnite element schemes in both the time- and frequency-domains. Herein, we also extend the applicability of elliptically-shaped truncation boundaries to semi-inﬁnite acoustic media. Numerical experiments for transient and time harmonic cases attest to the computational savings realized when elongated scatterers are surrounded by elliptically- or ellipsoidally-shaped boundaries, as opposed to the more commonly used circular or spherical truncation geome tries in either the full- or half-space cases (near-surface scatterers). Lastly, we treat the two-dimensional elastodynamics case based on a Helmholtz decomposition of the displacement vector ﬁeld. The decomposi tion allows for scalar wave equations to be written for the scalar and vector potential components. Thus, absorbing conditions similar to the ones writ ten for acoustics can be used for the elastodynamics case. The stability of the elastodynamics conditions for time-domain applications remains an open question.Show more Item Well-posedness for the space-time monopole equation and Ward wave map(2008-05) Czubak, Magdalena, 1977-; Uhlenbeck, Karen K.Show more We study local well-posedness of the Cauchy problem for two geometric wave equations that can be derived from Anti-Self-Dual Yang Mills equations on R2+2. These are the space-time Monopole Equation and the Ward Wave Map. The equations can be formulated in different ways. For the formulations we use, we establish local well-posedness results, which are sharp using the iteration methods.Show more